Combinatorial interpretation: a(n,k) counts the sets of lists (ordered subsets) obtained from partitioning the set {1..n}, with the lengths of the lists given by the k-th partition of n in A-St order. E.g. a(5,5) is computed from the number of sets of lists of lengths [1^1,2^2] (5th partition of 5 in A-St order). Hence a(5,5) = binomial(5,2)*binomial(3,2) = 5!/(1!*2!) = 60 from partitioning the numbers 1,2,...,5 into sets of lists of the type {[.],[..],[..]}.

This array, called M_3(2), is the k=2 member of a family of partition arrays generalizing A036040 which appears as M_3 = M_3(k=1). S2(2) = A105278 (unsigned Lah number triangle) is related to M_3(2) in the same way as S2(1), the Stirling2 number triangle, is related to M_3(1). - Wolfdieter Lang, Oct 19 2007

Another combinatorial interpretation: a(n,k) enumerates unordered forests of increasing binary trees which are described by the k-th partition of n in the Abramowitz-Stegun order. - Wolfdieter Lang, Oct 19 2007

A relation between partition polynomials formed from these "refined Lah numbers" and Lagrange inversion for an o.g.f. is presented in the link "Lagrange a la Lah" along with an e.g.f. and an umbral binary operator tree representation. - Tom Copeland, Apr 12 2011

The row partition polynomials of this array P(n,x_1,x_2,..,x_n), given in the Lang link, are n! * S(n,x_1,x_2,..,x_n), where S(n,x_1,..,x_n) are the elementary Schur polynomials, for which d/d(x_m) S(n,x_1,..,x_n) = S(n-m,x_1,..,x_(n-m)) with S(k,..) = 0 for k < 0, so d/d(x_m) P(n,x_1,..,x_n) = (n!/(n-m)!) P(n-m,x_1,..,x_(n-m)), confirming that the row poynomials form an Appell sequence in the indeterminate x_1 with P(0,..) = 1. See pg. 127 of the Ernst paper for more on these Schur polynomials.

The indeterminates of the partition polynomials can also be extracted using the Faber polynomials of A263916 with -n * x_n = F(n,S(1,x_1),..,S(n,x_1,..,x_n)) = F(n,P(1,x_1),..,P(n,x_1,..,x_n)/n!). Compare with A263634.

Also P(n,x_1,..,x_n) = ST1(n,x_1,2*x_2,..,n*x_n), where ST1(n,..) are the row partition polynomials of A036039.

(End)

EXAMPLE

Triangle starts:

[1];

[2,1];

[6,6,1];

[24,24,12,12,1];

[120,120,120,60,60,20,1];

...

a(5,6) = 20 = 5!/(3!*1!) because the 6th partition of 5 in A-St order is [1^3,2^1].